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The NLO contributions to the scalar pion form factors and the annihilation corrections to the decays
In this paper, by employing the factorization theorem, we made the
first calculation for the space-like scalar pion form factor at
the leading order (LO) and the next-to-leading order (NLO) level, and then
found the time-like scalar pion form factor by analytic
continuation from the space-like one. From the analytical evaluations and the
numerical results, we found the following points: (a) the NLO correction to the
space-like scalar pion form factor has an opposite sign with the LO one but is
very small in magnitude, can produce at most decrease to LO result in
the considered region; (b) the NLO time-like scalar pion form factor
describes the contribution to the
factorizable annihilation diagrams of the considered decays,
i.e. the NLO annihilation correction; (c) the NLO part of the form factor
is very small in size, and is almost independent with the
variation of cutoff scale , but this form factor has a large strong
phase around and may play an important role in producing large CP
violation for decays; and (d) for and decays, the newly known NLO annihilation correction can produce
only a very small enhancement to their branching ratios, less than in
magnitude, and therefore we could not interpret the well-known -puzzle
by the inclusion of this NLO correction to the factorizable annihilation
diagrams.Comment: 26 pages, 12 figures, 1 Table; Minor correction
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